Since the algorithm in [1]} works with directed graphs, we replace each undirected edge of \(G^\textsf {p}\)  into two edges
directed in the opposite directions. Also, we consider each walk \(W_i\)  of \(\mathcal {W}\)  as directed from \(s_i\)  to \(t_i\) .
Let \(\mathcal {F}\)  be the set of the faces of \(G^\textsf {p}\) . Note that there is a face of \(\mathcal {F}\)  whose boundary consists of
the copies of the edges of the innermost face \(F^*\)  of \(G\) . Recall that all frames contain the innermost face \(F^*\)  in their interiors.
To make the description easier, we also use \(F^*\)  to denote the face of \(G^\textsf {p}\)  corresponding to the innermost face of \(G\) .
